Automatica 46 (2010) 271–278 Contents lists available at ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica Unknown input observer for state affine systems: A necessary and sufficient condition ✩ Hassan Hammouri a,b,∗ , Zied Tmar c,d a Université de Lyon, F-69003, France b Université Lyon 1, CNRS UMR 5007 LAGEP (Laboratory of Process Control and Chemical Engineering), 43 bd du 11 novembre, 69100 Villeurbanne, France c Université de Tunis, Univrsité 7 Novembre à Carthage, Tunisia d LECAP (Study Laboratory of Automatic Process Control), Ecole Polytechnique de Tunisie, BP743, 2078 La Marsa, Tunisia article info Article history: Received 15 February 2008 Received in revised form 28 July 2009 Accepted 29 October 2009 Available online 5 December 2009 Keywords: Nonlinear systems Unknown input observer State affine models abstract The problem of designing an unknown input observer for linear systems and its application to fault detection is widely studied in the literature. For nonlinear systems, only subclasses of nonlinear systems and sufficient conditions have been stated. In this paper an unknown input observer design for state affine systems is considered. Based on the geometric approach, a necessary and sufficient condition is given for the existence of an unknown input observer. © 2009 Elsevier Ltd. All rights reserved. 1. Introduction An unknown input observer (UIO) is a dynamical system which estimates the unknown state (or a part) of a given system, in- dependently on the unknown inputs. One of the most successful robust observer design methods resorts to the disturbance decou- pling principle. The problem of UIO has been initiated by Basile and Marro (1969) and Guidorzi and Marro (1971). Since then, several contributions for designing reduced order and full order UIOs have been proposed: the geometric approach by Bhattacharyya (1978); a concept of strong detectability and its use in the UIO construc- tion has been introduced by Hautus (1983); the inversion algo- rithm, and the algebraic approaches, see for instance, Darouach, Zasadzinski, and Xu (1994) and Hou and Muller (1992). Observers for systems with unknown inputs play an essential role in robust model based fault detection. Initiated by Watanabe and Himmel- blau (1982), this problem has been extended to the detection of both sensor and actuator faults by Patton and Chen (1993) and ✩ This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Antonio Loria under the direction of Editor Andrew R. Teel. ∗ Corresponding address: Université Claude Bernard, Bat.308 G, 43 Bd Du 11 Novembre 1918, 69622 Villeurbanne Cedex, France. Tel.: +33 72431895; fax: +33 72431682. E-mail addresses: hammouri@lagep.univ-lyon1.fr (H. Hammouri), ziedtmar@ept.rnu.tn (Z. Tmar). Wunnenberg and Frank (1982). The fault detection and isola- tion based linear observer consists in designing an observer, called residual filter, generating an output which becomes sensitive to some faults and possibly insensitive to another one (FDI). Using the geometric approach, a solution has been proposed by Massoumnia, Verghese, and Willsky (1989) (the fundamental problem of resid- ual generation, FPRG). In the nonlinear context, many contribu- tions have been proposed in the literature. Initiated by Seliger and Frank (1991), the FDI for nonlinear systems was extended in a se- ries of papers: Hammouri, Kinnaert, and El Yaagoubi (1998, 1999) gave a sufficient condition for the existence of a solution of the FDI problem for a class of nonlinear systems. Then, through a differ- ential geometric approach, sufficient conditions for the existence of a solution of the FDI problem has been proposed successively by Hammouri, Kabore, and Kinnaert (2000) for state affine systems, and by De Persis and Isidori (2001) for control affine nonlinear sys- tems. For bilinear systems, Hammouri, Kabore, and Kinnaert (2001) gave a necessary and a sufficient condition for the existence of a solution of the FPRG. For LPV systems, a solution for the FPRG has been proposed by Bokor and Balas (2004). Using the inversion tech- niques based on the disturbance decoupling method, Edelmayer, Bokor, Szabo, and Szigeti (2004) proposed a solution for the FDI problem. Besides the geometric approach, many techniques which allow us to solve the FDI problem exist: (i) convex optimization procedures and LMI techniques, see for instance Zasadzinski, Ma- garotto, Rafaralahy, and Souley Ali (2003); (ii) Sliding mode ob- servers by Chen and Saif (2007), and Floquet, Barbot, Perruquetti, and Djemai (2004). 0005-1098/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2009.11.004